Question

Let $\varphi \left(x\right)={\left(f\right(x\left)\right)}^3-3{\left(f\right(x\left)\right)}^2+4f\left(x\right)+5x+3\sin x+4\cos x,\forall x\in R,$ where $f\left(x\right)$ is a differentiable function $\forall x\in R$, then

• A. $\varphi$ is increasing where $f$ is increasing
• B. $\varphi$ is decreasing where $f$ is decreasing
• C. $\varphi$ is decreasing whenever if $f^{\prime }\left(x\right)=-11$
• D. $\varphi$ is increasing whenever $f$ is decreasing

Answer 1

Answer: (A), (C)

$\varphi$ is decreasing whenever if $f^{\prime }\left(x\right)=-11$

${\varphi }^{\prime }\left(x\right)=f^{\prime }\left(x\right)\left(3f^2\right(x)-6f(x)+4)+5+3\cos x-4\sin x$
as $-5\le (3\cos x-4\sin x)\le 5$
So, $0\le (5+3\cos x-4\sin x)\le 10$
Whereas, $\left(3f^2\right(x)-6f(x)+4)$ is a polynomial in $f\left(x\right)$ whose minimum value $=\frac{48-36}{12}=1$
So, $f^{\prime }\left(x\right)>0$ $\Rightarrow {\varphi }^{\prime }\left(x\right)>0$
and $f^{\prime }\left(x\right)=-11$ $\Rightarrow {\varphi }^{\prime }\left(x\right)=-11\left(1\right)+10=-1$
$\therefore {\varphi }^{\prime }\left(x\right)<0$
$\Rightarrow \varphi \left(x\right)$ is decreasing.